Jensen's Inequality

Jensen's inequality: If $f$ is convex and $X$ is a random variable: $$f(E[X]) \leq E[f(X)]$$ If $f$ is concave, the inequality flips: $f(E[X]) \geq E[f(X)]$. Intuition: A convex function "bows up." The function value at the average is less than the average of the function values. Common applications: - $f(x) = x^2$ is convex: $E[X]^2 \leq E[X^2]$, so $\text{Var}(X) \geq 0$ - $f(x) = \ln x$ is concave: $\ln E[X] \geq E[\ln X]$, proving AM $\geq$ GM - $f(x) = e^x$ is convex: $e^{E[X]} \leq E[e^X]$ When to use: Bounding expectations of nonlinear functions. Proving that averages behave better than individual values. Establishing moment inequalities. This is a fundamental technique — many specific inequalities (AM-GM, power mean) are special cases.